|
Two's complement is a mathematical operation on binary numbers, as well as a binary signed number representation based on this operation. Its wide use in computing makes it the most important example of a radix complement. The two's complement of an -bit number is defined as the complement with respect to ; in other words, it is the result of subtracting the number from , which in binary is one followed by zeroes. This is also equivalent to taking the ones' complement and then adding one, since the sum of a number and its ones' complement is all 1 bits. The two's complement of a number behaves like the negative of the original number in most arithmetic, and positive and negative numbers can coexist in a natural way. In two's-complement representation, positive numbers are simply represented as themselves, and negative numbers are represented by the two's complement of their absolute value;〔David J. Lilja and Sachin S. Sapatnekar, ''Designing Digital Computer Systems with Verilog'', Cambridge University Press, 2005 (online )〕 two tables on the right provide examples for = 3 and = 8. In general, negation (reversing the sign) is performed by taking the two's complement. This system is the most common method of representing signed integers on computers.〔E.g. "Signed integers are two's complement binary values that can be used to represent both positive and negative integer values.", Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual, Volume 1: Basic Architecture, November 2006〕 An -bit two's-complement numeral system can represent every integer in the range to while ones' complement can only represent integers in the range to . The two's-complement system has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers (as long as the inputs are represented in the same number of bits and any overflow beyond those bits is discarded from the result). This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, obviating the subtleties associated with negative zero, which exists in ones'-complement systems. ==History== The method of complements had long been used to perform subtraction in decimal adding machines and mechanical calculators. John von Neumann suggested use of two's complement binary representation in his 1945 ''First Draft of a Report on the EDVAC'' proposal for an electronic stored-program digital computer.〔 〕 The 1949 EDSAC, which was inspired by the ''First Draft'', used two's complement representation of binary numbers. Many early computers, including the CDC 6600, use ones' complement notation. The IBM 700/7000 series scientific machines use sign/magnitude notation, except for the index registers which are two's complement. Early commercial two's complement computers include the Digital Equipment Corporation PDP-5 and the 1963 PDP-6. The System/360, introduced in 1964 by IBM, then the dominant player in the computer industry, made two's complement the most widely used binary representation in the computer industry. The first minicomputer, the PDP-8 introduced in 1965, uses two's complement arithmetic as do the 1969 Data General Nova, the 1970 PDP-11, and almost all subsequent minicomputers and microcomputers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Two's complement」の詳細全文を読む スポンサード リンク
|